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Surface Areas and Volumes - Volume of a Combination of Solids

Grade 10CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A combination of solids refers to a three-dimensional object formed by joining two or more basic geometric shapes, such as a cone placed on top of a hemisphere or a cylinder with hemispherical ends. Visually, these objects look like single, continuous structures where different geometries meet at shared boundaries like circles or squares.

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The total volume of a combined solid is the sum of the volumes of its constituent parts. Visually, you can imagine this as the total amount of space occupied by each part separately; for example, if a solid is made of a cylinder and a cone, its total volume is simply Vtotal=Vcylinder+VconeV_{\text{total}} = V_{\text{cylinder}} + V_{\text{cone}}.

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Common Interface: When two solids are joined at a base, such as a cone on a cylinder, they typically share the same radius rr. In a diagram, this appears as a perfectly aligned circular edge where the curved surface of the cone meets the flat circular top of the cylinder.

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Volume of Subtractive Solids: When a shape is 'carved out' or removed from another solid (like a conical cavity drilled into a wooden cube), the volume of the remaining solid is calculated by subtracting the volume of the removed part from the original solid's volume. Visually, this is represented by an empty space or 'hole' inside the primary shape.

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Conservation of Volume: When a solid of one shape is melted and recast into another shape (for example, melting several small spheres to form one large cylinder), the total volume of the material remains constant. This is modeled by the equation Voriginal=VnewV_{\text{original}} = V_{\text{new}}. Visually, the amount of 'stuff' or matter remains the same even though the outer boundaries change completely.

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Component Heights: In complex solids like a capsule (a cylinder with two hemispherical ends), the 'total length' is the sum of the cylinder's height and the radii of the two ends. Visually, to find the cylinder's height (hh), you must look at the total length and subtract the radius (rr) from both ends: h=Total Length−2rh = \text{Total Length} - 2r.

📐Formulae

Volume of a Cuboid: V=l×b×hV = l \times b \times h

Volume of a Cube: V=a3V = a^3

Volume of a Right Circular Cylinder: V=πr2hV = \pi r^2 h

Volume of a Right Circular Cone: V=13πr2hV = \frac{1}{3} \pi r^2 h

Volume of a Sphere: V=43πr3V = \frac{4}{3} \pi r^3

Volume of a Hemisphere: V=23πr3V = \frac{2}{3} \pi r^3

💡Examples

Problem 1:

A solid is in the form of a cone standing on a hemisphere with both their radii being equal to 1 cm1 \text{ cm} and the height of the cone is equal to its radius. Find the volume of the solid in terms of π\pi.

Solution:

Step 1: Identify the two solids. The object is a combination of a cone and a hemisphere. \nStep 2: List the given dimensions. Radius (rr) of both cone and hemisphere = 1 cm1 \text{ cm}. Height (hh) of the cone = 1 cm1 \text{ cm}. \nStep 3: Calculate the volume of the cone part: V1=13πr2h=13π(1)2(1)=13π cm3V_1 = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (1)^2 (1) = \frac{1}{3} \pi \text{ cm}^3. \nStep 4: Calculate the volume of the hemisphere part: V2=23πr3=23π(1)3=23π cm3V_2 = \frac{2}{3} \pi r^3 = \frac{2}{3} \pi (1)^3 = \frac{2}{3} \pi \text{ cm}^3. \nStep 5: Add the volumes for the total volume: V=V1+V2=13π+23π=33π=π cm3V = V_1 + V_2 = \frac{1}{3} \pi + \frac{2}{3} \pi = \frac{3}{3} \pi = \pi \text{ cm}^3.

Explanation:

Since the cone and hemisphere are joined together, we simply calculate their individual volumes using the standard formulae and add them. The problem asks for the answer 'in terms of π\pi', so we do not substitute 3.143.14 or 227\frac{22}{7}.

Problem 2:

A decorative block is made of two solids — a cube and a hemisphere. The base of the block is a cube with edge 5 cm5 \text{ cm}, and the hemisphere fixed on the top has a diameter of 4.2 cm4.2 \text{ cm}. Find the total volume of the block. (Use π=227\pi = \frac{22}{7})

Solution:

Step 1: Identify the parts. We have a cube and a hemisphere sitting on top of it. \nStep 2: Note the dimensions. Edge of cube (aa) = 5 cm5 \text{ cm}. Radius of hemisphere (rr) = 4.22=2.1 cm\frac{4.2}{2} = 2.1 \text{ cm}. \nStep 3: Calculate the volume of the cube: Vcube=a3=53=125 cm3V_{\text{cube}} = a^3 = 5^3 = 125 \text{ cm}^3. \nStep 4: Calculate the volume of the hemisphere: Vhemi=23πr3=23×227×(2.1)3V_{\text{hemi}} = \frac{2}{3} \pi r^3 = \frac{2}{3} \times \frac{22}{7} \times (2.1)^3. \nVhemi=23×227×9.261=19.404 cm3V_{\text{hemi}} = \frac{2}{3} \times \frac{22}{7} \times 9.261 = 19.404 \text{ cm}^3. \nStep 5: Total volume of the block = Vcube+Vhemi=125+19.404=144.404 cm3V_{\text{cube}} + V_{\text{hemi}} = 125 + 19.404 = 144.404 \text{ cm}^3.

Explanation:

In this problem, the hemisphere is placed on top of the cube. The volume of the block is the sum of the space occupied by the cube and the space occupied by the hemisphere. Note that for volume, the area where they touch (the base of the hemisphere) does not need to be subtracted, unlike when calculating surface area.